Quantum oscillations and quasilinear magnetoresistance in the topological semimetal candidate $\mathrm{Sc}{\mathrm{Sn}}_2$

Novel compounds with two-dimensional square lattices formed by group IV or V elements (Si, Sn, Ge, Bi, and Sb) have been attracting a great deal of attention in recent times, mainly because of the possible emergence of various topological phases therein. Here, we successfully grow the single crystals of Sn-square-net based material $\mathrm{Sc}{\mathrm{Sn}}_2$, and systematically perform their magnetization and magnetotransport measurements. Clear quantum oscillations emerge in the magnetization isotherms along different field orientations, from which nonzero Berry phases are extracted, implying that $\mathrm{Sc}{\mathrm{Sn}}_2$ harbors three-dimensional Fermi surfaces and nontrivial electronic states. Similar to many other topological semimetals with extremely large magnetoresistance, $\mathrm{Sc}{\mathrm{Sn}}_2$ shows field-induced resistivity enhancement as well, which has been proven to be not of a gap opening origin. Besides, at low temperature, large magnetoresistance with a quasilinear field dependence is observed. Our analysis of the magnetotransport data finds that the quasilinear magnetoresistance in $\mathrm{Sc}{\mathrm{Sn}}_2$ cannot be understood by several familiar mechanisms proposed in the literature. These findings are expected to have far reaching implications for both the fundamental understanding and magnetoresistance device application of topological semimetal materials.

Figure 1

(a) Crystallographic structure of $\mathrm{Sc}{\mathrm{Sn}}_2$, a trigonal prism-type Sc slab centered by Sn atoms and a Sn layer. (b) Photographs of synthesized single-crystal $\mathrm{Sc}{\mathrm{Sn}}_2$ placed on a millimeter grid. (c) Powder XRD results (black solid sphere) at room T. The red line is the best fit from the Rietveld refinement using fullprof software. Small amount of residual Sn flux (marked with a black asterisk) is present in the x-ray pattern (d) XRD patterns with the x rays along the perpendicular direction of crystal surfaces. (e) EDX for $\mathrm{Sc}{\mathrm{Sn}}_2$ crystals. (f) A SEM image for $\mathrm{Sc}{\mathrm{Sn}}_2$ crystals and elemental distributions of Sc and Sn acquired by scanning EDX. (g) $C_{\mathrm{p}}$ vs T and $C_{\mathrm{p}}/T$ vs T (inset) plots for $\mathrm{Sc}{\mathrm{Sn}}_2$ crystals.

Figure 2

(a) Electronic band structures for $\mathrm{Sc}{\mathrm{Sn}}_2$ without SOC, in which $P_1-P_6$ denote the six crossing points near the FL, respectively. (b) Positions of the nodal loops in BZ. (c) Electronic band structures for $\mathrm{Sc}{\mathrm{Sn}}_2$ with SOC. Tiny gaps of 42, 0.8, 0.3, 17.1, 128, and 33 meV are opened for $P_1-P_6$, respectively. (d) Contributions from Sn orbitals to $P_3$.

Figure 3

(a),(b) M isotherms for $\mathrm{Sc}{\mathrm{Sn}}_2$ with $B\parallel a$ and $B\parallel ab$, respectively. (c),(d) ΔM vs 1/B for $B\parallel a$ and $B\parallel ab$, respectively. Insets show ΔM vs 1/B curves and their LK fittings at 1.8 K for $B\parallel c$ and $B\parallel ab$, respectively. (e),(f) FFT spectra of the out-of-plane (c) and in-plane (d) ΔM for $\mathrm{Sc}{\mathrm{Sn}}_2$ at various Ts.

Figure 4

(a), (b) T dependence of the out-of-plane ($B\parallel c$) and in-plane ($B\parallel ab$) normalized FFT amplitude for $\mathrm{Sc}{\mathrm{Sn}}_2$; the red solid lines represent the LK fit for ${m_{\alpha }}^{*}$, ${m_{\beta }}^{*}$, and ${m_{\gamma }}^{*}$. (c) $\mathrm{\Delta }{M}_{\alpha }$, $\mathrm{\Delta }{M}_{2\alpha }$, $\mathrm{\Delta }{M}_{\alpha }+\mathrm{\Delta }{M}_{2\alpha }$, and the experimental data ΔM for $B\parallel c$ vs 1/B. (d) $\mathrm{\Delta }{M}_{\beta }$, $\mathrm{\Delta }{M}_{\gamma }$, $\mathrm{\Delta }{M}_{\beta }+\mathrm{\Delta }{M}_{\gamma }$, and the experimental data ΔM for $B\parallel ab$ vs 1/B. (e), (f) LL index fan diagram for $F_{\alpha }$, $F_{\beta }$, and $F_{\gamma }$. N − 1/4 and N − 3/4 (N, integer LL indices) are assigned to ΔM oscillation minima and maxima, respectively.

Figure 5

(a) Low-T ${\rho }_{\mathrm{xx}}$ and its fit with ${\rho }_{\mathrm{xx}}={\rho }_0+a{\mathrm{x}}^{\mathrm{n}}$ (${\rho }_0=5.22\mu \mathrm{\Omega }\mathrm{cm}$, $a=0.03\mu \mathrm{\Omega }\mathrm{cm}{\mathrm{K}}^{\text{–}n}$, and $n=1.98$). The upper left inset shows ${\rho }_{\mathrm{xx}}$ as a function of T from 2 to 300 K, where the blue line denotes a linear T dependence. The lower right inset displays the actual device for ${\rho }_{\mathrm{xx}}$ measurements. (b) ${\rho }_{\mathrm{xx}}$ vs T between 2 and 60 K under different Bs. The inset shows normalized MR [MR(T)/MR(2 K)] as a function of T under various Bs. (c), (d) MR vs B at different Ts. (e) A (left axis) and $n$ (right axis) as a function of T. Panel (f) and its inset: Kohler's plot at different T using $\mathrm{MR}\sim {\left[B/\rho \left(0\right)\right]}^{1.3}$ for $2\mathrm{K}<T<150\mathrm{K}$ and $\mathrm{MR}\sim {\left[B/\rho \left(0\right)\right]}^{2.24}$ for $200\mathrm{K}<T<300\mathrm{K}$.

Figure 6

(a) ${\rho }_{\mathrm{xy}}$ as a function of B at different T for $\mathrm{Sc}{\mathrm{Sn}}_2$. Inset shows the actual device for ${\rho }_{\mathrm{xy}}$ measurements. (b) Evolution of carrier concentrations ($n_{\mathrm{h}}$ and $n_{\mathrm{e}}$) and mobilities (${\mu }_{\mathrm{h}}$ and ${\mu }_{\mathrm{e}}$) as a function of T in $\mathrm{Sc}{\mathrm{Sn}}_2$. (c) High-B $d\mathrm{MR}/dB$ (left axis) and μ (right axis) vs T. Inset plots: ${\mu }_{\mathrm{h}}$ and ${\mu }_{\mathrm{e}}$ as a function of $d\mathrm{MR}/dB$. (d) $B_{\mathrm{c}}$ vs ${\mu }^{\text{–}1}$. Inset shows the definition of $B_{\mathrm{c}}$ at 2 K.